# Characteristics And Features of Mathematics

**The key features and characteristics of mathematics are:**

- Logical sequence
- Applicability
- Mathematical systems
- Generalization and classification
- Structure
- Mathematical Language and Symbolism
- Rigor and logic
- Abstractness
- Precision and accuracy

## 1. Logical Sequence

**The earliest mathematics was firmly empirical (experiential), rooted in man’s perception of:**

- Number (
*quantity*) - Space (
*configuration*) - Time, and
- Change (
*transformation*)

Today mathematical fact can be established without reference to empirical reality. **All this transpired with the gradual process of**

*Experience,**Abstraction, and**Generalization in the field of mathematics.*

**Mathematics is, today, built upon abstract concepts whose relationship with real experiences is useful but not essential. **These abstractions mean that mathematical fact is certainly be influenced by reality, as it often is, but it is not considered mathematical fact until it is established according to the logical requirements of modern mathematics.

It is this evolution from empirical science to axiomatic science that has established derivability as the basis for mathematics and made logical sequence a main feature of mathematics.

## 2. Applicability

Concepts and principles become more functional and meaningful only when they are related to actual practical applications. **It is the natural instinct of man to**

- Seek explanation,
- To generalize, and
- To attempt to improve the organization of his knowledge.

“Knowledge is power only when it is applied”.

**Whenever knowledge is applied, especially, related to daily life situations it makes the learning of any discipline more meaningful and significant.** Mathematical truth turns are applicable in very distinct areas of application from across the universe to across the street.

__The study of mathematics requires the learner to apply the skills acquired to new situations.__

## 3. Mathematical systems

**A typical mathematical system has the following four parts:**

- Undefined terms,
- Defined terms,
- Axioms and
- Theorems.

### a. Undefined terms

**In geometry or in any other mathematical system, we have to start with some terms,** these terms are *typically extremely simple and basic objects*, so simple that they resist being described in terms of simpler objects.

**Example:** *point, line, set, variable, plane, etc.*

The choice of the undefined terms is completely arbitrary and generally facilitates the development of the structure.

### b. Defined terms

**For example**, A triangle having 3 equal sides is an equilateral triangle. Thus to define an equilateral triangle, one should have learned the terms

*Triangle,**Equal, or**Sides.*

### c. Axioms

Axioms or postulates are a statement in a mathematical system that describes the relationships existing among the undefined terms of the system.**For example: To describe the relationship existing among undefined terms ‘line’ and ‘point’ some axioms that can be used are:**

*There can be one and only straight line joining two points.**Two lines meet at a point.**A line has one and only one mid-point.*

### d. Theorems

A statement that we arrive at by successive application of the rule of implication to the axioms and statements previously arrived at is called a theorem.

**For instance: The rule of implication states that**

*the statement p implies the statement q and**If the statement p is true, then the statement q will be true.*

When we apply the rule of application to the axioms we generate new statements. Again we may apply this rule to these new statements.

## 4. Generalization and classification

The generalization and classification of mathematics are very straightforward in contrast to others fields of thought and activity. Mathematics unites numerous findings, conclusions, assumptions, etc. under one head, and from that makes schematic arrangements and classifications.

**Some of the examples of successive generalizations in mathematics are:**

- Number concept has itself widened from that of the whole number when it included successively negative numbers, fractional numbers, imaginary numbers, and irrational numbers.
- One of the significant traits of algebra is its generalized handling of the processes of arithmetic.
- In geometry, there are frequent occurrences for grouping and generating results.

When the students evolve there own concepts, theorems, definitions he/she is making generalizations.

## 5. Structure

A mathematical structure is a mathematical system with one or more explicitly recognized (mathematical) properties.**Generally speaking, a structure denotes ‘the formation, arrangement, and articulation of parts in anything build-up by nature or art’.For example:**

- If ‘S’ is a non-empty set on which one or more operations have been uniquely defined with respect to an equivalence relation, then the set S together with the operation(s) is called a mathematical system.
- Using one or more of the mathematical systems like commutative, associative, or distributive properties we may create a structure.

**The mathematical structure has a variety of arrangements, formations, which results in putting parts together. For instance:**

- A structure that comprises of a mathematical system <S; O> with one operation, in which the operation O is associative is called a semi-group.

Thus mathematics has got definite logical structures. These structures * ensure the order and beauty of mathematics*.

## 6. Mathematical Language and Symbolism

Over the course of the past 3000 years, mankind has created sophisticated spoken and written natural languages which are tremendously efficient for expressing a variety of meanings, moods, and motives.

*Man has the ability to assign symbols for ideas and objects.*

**The language in which Mathematics is developed is no less, and, when mastered, provides a highly effective and powerful tool for**

*Mathematical Expression,**Exploration,**Reconstruction After Exploration, and**Communication.*

*Usage of symbols constructs mathematics language and makes it more elegant and precise than any other language.* **Mathematical language and symbols**

- Cut short the lengthy statements.
- Help the expression of ideas or things in the exact form.
- Mathematical language is free from verbosity.
- Mathematical symbols help to form and clear exact expression of facts.

*All mathematical operations, relations, statements are expressed using mathematical symbols. **T*he training that mathematics provides in the use of symbols is excellent preparation for other sciences. For example:

- We can state the commutative law of addition and multiplication in a real number system in the verbal form as: ‘the addition and multiplication of two real numbers is independent of the order in which they are combined’.
- In concise form as:
**a + b = b + a**(addition), and**a * b = b * a**(multiplication).

“Mathematics is the language of physical sciences and certainly no more marvelous language was ever created by the mind of man”. - Lindsay

## 7. Rigor and logic

Logic is essential in mathematics; logic regulates the pattern of deductive proof through which mathematics is developed. **In modern times;**

*The primary pedagogical objective of Mathematics*is that it must be understood institutively in geometrical or physical terms.*The secondary pedagogical objective of Mathematics*is its rigorous presentation.

“Argument concludes a question, but it does not make us feel certain, or acquiesce in the contemplation of truth, except the truth also be found to be so by experience”- AS Roger Bacon

## 8. Abstractness

Everything in math cannot be learned through experiences with concrete objects the same way as other disciplines. Some mathematical concepts can be learned only through their definition and they may not have a physical matching part to be extracted from.

**Mathematics is abstract in the sense that mathematics does not deal with actual objects in much the same way as physics. But, in fact, mathematics questions, as a rule, cannot be settled by direct appeal to experiment.**

**For instance: **Our whole thinking is based on the belief or assumption that there are infinitely many numbers, there are infinitely many fractions between 0 and 1and therefore, counting never stops.

**Infinity is something that we can never experience and yet it is a central concept of mathematics.**

Man has no way of knowing, calculating, and justifying this as a man cannot observe and count all these which makes, Infinity, abstract concept, as it is not a concept corresponding to any object that man has seen or is likely to see.

**Some other examples of abstract concepts in math are:**

*Prime numbers,**Probability,**Limit and function,**Continuous functions etc.*

These all are abstract in the sense they can be learned only through their definitions and it is not possible to provide concrete objects to correspond to such concepts.

**Even some of the concepts which one argues to be concrete are also abstract. For example:**

- Concepts such as a line, a diagonal, a point, a circle, a ray, etc., which seen as concepts that are learned through observation of concrete instances, and as a result, they are concrete. But a figure of a circle, a dot (point), a line drawn on a board, are all mere representations of the concepts and they are not objects themselves.

## 9. Precision and accuracy

*Mathematics is known as an ***‘exact’*** science because of its precision*. It is perhaps the only subject that can claim certainty of results. **Even when there is an emphasis on approximation, mathematical results have some degree of accuracy.**

- There is no midway possible in Mathematics. Mathematics is either correct or incorrect, right or wrong, it is accepted or rejected.
- Mathematics can decide whether or not its conclusions are right.
- Mathematicians can verify the validity of the results and convince others of their validity with consistency and objectivity.
- Mathematics true or false holds for everyone who uses mathematics, at any level, not only for the expert.

Mathematical culture is that what you say should be correct. What you say should have a definition. You should know the definition and limits of what you are stating, claiming, or saying.

**Thus, the modern mathematical culture of precision arises because:**

- Mathematics has developed a highly symbolic and precise language.
- Mathematical concepts have developed in a dialectic manner that allows for the adjustment, adaptation, and cumulative refinement of concepts based on experiences.
- Mathematical reasoning is expected to be correct.

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